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Problem Books in Mathematics

Series Editor

Peter Winkler

Department of Mathematics

Dartmouth College

Hanover, NH 03755

USA

[email protected]

For other titles published in the series, go to

www.springer.com/series/714

Vladimir Jankovic Nikola Petrovic

Dusan Djukic Ivan Matic

The IMO Compendium

Second Edition

A Collection of Problems Suggested for The International Mathematical Olympiads: 1959-2009

Springer New York Dordrecht Heidelberg London

Springer Science+Business Media, LLC 2011

subject to proprietary rights.

Printed on acid-free paper

Springer is part of Springer Science+Business Media (www.springer.com)

Duan Djuki Department of Mathematics University of Toronto

Canada [email protected]

Ivan Mati Department of Mathematics Duke University

USA

Vladimir Jankovi Department of Mathematics University of Belgrade Studentski Trg 16 11000 Belgrade Serbia

Nikola Petrovi Science Department Texas A&M University PO Box 23874 Doha Qatar

ISSN 0941-3502ISBN 978-1-4419-9853-8 e-ISBN 978-1-4419-9854-5

or by similar or dissimilar methodology now known or hereafter developed is forbidden.

DOI 10.1007/978-1-4419-9854-5

[email protected] [email protected]

Durham, North Carolina 27708

Library of Congress Control Number: 2011926996

All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software,

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Preface

The International Mathematical Olympiad (IMO) exists for more than 50 years andhas already created a very rich legacy and firmly established itself as the most presti-gious mathematical competition in which a high-school student could aspire to par-ticipate. Apart from the opportunity to tackle interesting and very challenging math-ematical problems, the IMO represents a great opportunity for high-school studentsto see how they measure up against students from the rest of the world. Perhaps evenmore importantly, it is an opportunity to make friends and socialize with studentswho have similar interests, possibly even to become acquainted with their future col-leagues on this first leg of their journey into the world of professional and scientificmathematics. Above all, however pleasing or disappointing the final score may be,preparing for an IMO and participating in one is an adventure that will undoubtedlylinger in ones memory for the rest of ones life. It is to the high-school-aged aspiringmathematician and IMO participant that we devote this entire book.

The goal of this book is to include all problems ever shortlisted for the IMOs ina single volume. Up to this point, only scattered manuscripts traded among differentteams have been available, and a number of manuscripts were lost for many years orunavailable to many.

In this book, all manuscripts have been collected into a single compendium ofmathematics problems of the kind that usually appear on the IMOs. Therefore, webelieve that this book will be the definitive and authoritative source for high-schoolstudents preparing for the IMO, and we suspect that it will be of particular benefit incountries lacking adequate preparation literature. A high-school student could spendan enjoyable year going through the numerous problems and novel ideas presentedin the solutions and emerge ready to tackle even the most difficult problems on anIMO. In addition, the skill acquired in the process of successfully attacking difficultmathematics problems will prove to be invaluable in a serious and prosperous careerin mathematics.

However, we must caution our aspiring IMO participant on the use of this book.Any book of problems, no matter how large, quickly depletes itself if the readermerely glances at a problem and then five minutes later, having determined that theproblem seems unsolvable, glances at the solution.

VI Preface

The authors therefore propose the following plan for working through the book.Each problem is to be attempted at least half an hour before the reader looks atthe solution. The reader is strongly encouraged to keep trying to solve the problemwithout looking at the solution as long as he or she is coming up with fresh ideasand possibilities for solving the problem. Only after all venues seem to have beenexhausted is the reader to look at the solution, and then only in order to study itin close detail, carefully noting any previously unseen ideas or methods used. Tocondense the subject matter of this already very large book, most solutions havebeen streamlined, omitting obvious derivations and algebraic manipulations. Thus,reading the solutions requires a certain mathematical maturity, and in any case, thesolutions, especially in geometry, are intended to be followed through with penciland paper, the reader filling in all the omitted details. We highly recommend thatthe reader mark such unsolved problems and return to them in a few months to seewhether they can be solved this time without looking at the solutions. We believe thisto be the most efficient and systematic way (as with any book of problems) to raiseones level of skill and mathematical maturity.

We now leave our reader with final words of encouragement to persist in thisjourney even when the difficulties seem insurmountable and a sincere wish to thereader for all mathematical success one can hope to aspire to.

Belgrade, Duan DjukicNovember 2010 Vladimir Jankovic

Ivan MaticNikola Petrovic

Over the previous years we have created the website: www.imomath.com.There you can find the most current information regarding the book, the list of de-tected errors with corrections, and the results from the previous olympiads. This sitealso contains problems from other competitions and olympiads, and a collection oftraining materials for students preparing for competitions.

We are aware that this book may still contain errors. If you find any, please notifyus at [email protected]. If you have any questions, comments, or sugges-tions regarding both our book and our website, please do not hesitate to write to usat the above email address. We would be more than happy to hear from you.

Preface VII

Acknowledgements

The making of this book would have never been possible without the help of numer-ous individuals, whom we wish to thank.

First and foremost, obtaining manuscripts containing suggestions for IMOs wasvital in order for us to provide the most complete listing of problems possible. We ob-tained manuscripts for many of the years from the former and current IMO team lead-ers of Yugoslavia / Serbia, who carefully preserved these valuable papers throughoutthe years. Special thanks are due to Prof. Vladimir Micic, for some of the oldestmanuscripts, and to Prof. Zoran Kadelburg. We also thank Prof. Djordje Dugoijaand Prof. Pavle Mladenovic. In collecting shortlisted and longlisted problems wewere also assisted by Prof. Ioan Tomescu from Romania, H Duy Hng from Viet-nam, and Zhaoli from China.

A lot of work was invested in cleaning up our giant manuscript of errors. Specialthanks in this respect go to David Kramer, our copy-editor, and to Prof. Titu An-dreescu and his group for checking, in great detail, the validity of the solutions inthis manuscript, and for their proposed corrections and alternative solutions to sev-eral problems. We also thank Prof. Abderrahim Ouardini from France for sendingus the list of countries of origin for the shortlisted problems of 1998, Prof. DorinAndrica for helping us compile the list of books for reference, and Prof. LjubomirCukic for proofreading part of the manuscript and helping us correct several errors.

We would also like to express our thanks to all anonymous authors of the IMOproblems. Without them, the IMO would obviously not be what it is today. It is apity that authors names are not registered together with their proposed problems. Inan attempt to change this, we have tried to trace down the authors of the problems,with partial success. We are thankful to all people who were so kind to help us inour investigation. The names we have found so far are listed in Appendix C. In manycases, the original solutions of the authors were used, and we duly acknowledge thisimmense contribution to our book, though once again, we regret that we cannot dothis individually. In the same vein, we also thank all the students participating in theIMOs, since we have also included some of their original solutions in this book.

We thank the following individuals who discussed problems with us and helpedus with correcting the mistakes from the previous edition of the book: Xiaomin Chen,Orlando Dhring, Marija Jelic, Rudolfs Kreicbergs, Stefan Mehner, Yasser AhmadyPhoulady, Dominic Shau Chin, Juan Ignacio Restrepo, Arkadii Slinko, Harun iljak,Josef Tkadlec, Ilan Vardi, Gerhard Woeginger, and Yufei Zhao.

The illustrations of geometry problems were done in WinGCLC, a program cre-ated by Prof. Predrag Janicic. This program is specifically designed for creating geo-metric pictures of unparalleled complexity quickly and efficiently. Even though it isstill in its testing phase, its capabilities and utility are already remarkable and worthyof highest compliment.

Finally, we would like to thank our families for all their love and support duringthe making of this book.

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The International Mathematical Olympiad . . . . . . . . . . . . . . . . . . . . . . 11.2 The IMO Compendium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 Basic Concepts and Facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.1.2 Recurrence Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.1.3 Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.1.4 Groups and Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3 Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.3.1 Triangle Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.3.2 Vectors in Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3.3 Barycenters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.4 Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.3.5 Circle Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3.6 Inversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.3.7 Geometric Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.8 Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3.9 Formulas in Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

2.4 Number Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4.1 Divisibility and Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . 192.4.2 Exponential Congruences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202.4.3 Quadratic Diophantine Equations . . . . . . . . . . . . . . . . . . . . . . . 212.4.4 Farey Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.5 Combinatorics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.5.1 Counting of Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222.5.2 Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

X Contents

3 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.1 IMO 1959 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 IMO 1960 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29







3.8.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353.8.2 Some Longlisted Problems 19591966 . . . . . . . . . . . . . . . . . . 35

3.9 IMO 1967 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.9.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.9.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.10 IMO 1968 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.10.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.10.2 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49


3.12 IMO 1970 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.12.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.12.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 623.12.3 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68





Contents XI

3.16.3 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 953.17 IMO 1975 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.17.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 973.17.2 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97











3.28 IMO 1987 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1923.28.1 Contest Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

XII Contents

3.28.2 Longlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1923.28.3 Shortlisted Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200














Contents XIII










4 Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3474.1 Contest Problems 1959 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3474.2 Contest Problems 1960 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3494.3 Contest Problems 1961 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3514.4 Contest Problems 1962 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3534.5 Contest Problems 1963 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3544.6 Contest Problems 1964 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3554.7 Contest Problems 1965 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3574.8 Contest Problems 1966 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3594.9 Longlisted Problems 1967 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3614.10 Shortlisted Problems 1968 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3744.11 Contest Problems 1969 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3804.12 Shortlisted Problems 1970 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3834.13 Shortlisted Problems 1971 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3894.14 Shortlisted Problems 1972 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3964.15 Shortlisted Problems 1973 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4014.16 Shortlisted Problems 1974 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 407

XIV Contents

4.17 Shortlisted Problems 1975 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4134.18 Shortlisted Problems 1976 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4184.19 Longlisted Problems 1977 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4224.20 Shortlisted Problems 1978 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4374.21 Shortlisted Problems 1979 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4454.22 Shortlisted Problems 1981 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4534.23 Shortlisted Problems 1982 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4614.24 Shortlisted Problems 1983 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4674.25 Shortlisted Problems 1984 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4764.26 Shortlisted Problems 1985 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4834.27 Shortlisted Problems 1986 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4914.28 Shortlisted Problems 1987 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4984.29 Shortlisted Problems 1988 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5084.30 Shortlisted Problems 1989 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5234.31 Shortlisted Problems 1990 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5374.32 Shortlisted Problems 1991 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5504.33 Shortlisted Problems 1992 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5634.34 Shortlisted Problems 1993 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5734.35 Shortlisted Problems 1994 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5854.36 Shortlisted Problems 1995 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5934.37 Shortlisted Problems 1996 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6064.38 Shortlisted Problems 1997 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6224.39 Shortlisted Problems 1998 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6364.40 Shortlisted Problems 1999 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6504.41 Shortlisted Problems 2000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6644.42 Shortlisted Problems 2001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6774.43 Shortlisted Problems 2002 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6914.44 Shortlisted Problems 2003 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7024.45 Shortlisted Problems 2004 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7154.46 Shortlisted Problems 2005 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7304.47 Shortlisted Problems 2006 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7424.48 Shortlisted Problems 2007 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7544.49 Shortlisted Problems 2008 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7654.50 Shortlisted Problems 2009 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 777

A Notation and Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791A.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 791A.2 Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 792

B Codes of the Countries of Origin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 795

C Authors of Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 797

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 805

1

Introduction

1.1 The International Mathematical Olympiad

The International Mathematical Olympiad (IMO) is the most important and presti-gious mathematical competition for high-school students. It has played a significantrole in generating wide interest in mathematics among high school students, as wellas identifying talent.

In the beginning, the IMO was a much smaller competition than it is today. In1959, the following seven countries gathered to compete in the first IMO: Bulgaria,Czechoslovakia, German Democratic Republic, Hungary, Poland, Romania, and theSoviet Union. Since then, the competition has been held annually. Gradually, otherEastern-block countries, countries from Western Europe, and ultimately numerouscountries from around the world and every continent joined in. (The only year inwhich the IMO was not held was 1980, when for financial reasons no one steppedin to host it. Today this is hardly a problem, and hosts are lined up several years inadvance.) In the 50th IMO, held in Bremen, no fewer than 104 countries took part.

The format of the competition quickly became stable and unchanging. Eachcountry may send up to six contestants and each contestant competes individually(without any help or collaboration). The country also sends a team leader, who par-ticipates in problem selection and is thus isolated from the rest of the team until theend of the competition, and a deputy leader, who looks after the contestants.

The IMO competition lasts two days. On each day students are given four and ahalf hours to solve three problems, for a total of six problems. The first problem isusually the easiest on each day and the last problem the hardest, though there havebeen many notable exceptions. ((IMO96-5) is one of the most difficult problemsfrom all the Olympiads, having been fully solved by only six students out of severalhundred!) Each problem is worth 7 points, making 42 points the maximum possiblescore. The number of points obtained by a contestant on each problem is the result ofintense negotiations and, ultimately, agreement among the problem coordinators, as-signed by the host country, and the team leader and deputy, who defend the interestsof their contestants. This system ensures a relatively objective grade that is seldomoff by more than two or three points.

1 Springer Science + Business Media, LLC 2011

et al., The IMO Compendium, Problem Books in Mathematics, DOI 10.1007/978-1-4419-9854-5_1,

D. Djuki

2 1 Introduction

Though countries naturally compare each others scores, only individual prizes,namely medals and honorable mentions, are awarded on the IMO. Fewer than onetwelfth of participants are awarded the gold medal, fewer than one fourth are awardedthe gold or silver medal, and fewer than one half are awarded the gold, silver orbronze medal. Among the students not awarded a medal, those who score 7 points onat least one problem are awarded an honorable mention. This system of determiningawards works rather well. It ensures, on the one hand, strict criteria and appropriaterecognition for each level of performance, giving every contestant something to strivefor. On the other hand, it also ensures a good degree of generosity that does notgreatly depend on the variable difficulty of the problems proposed.

According to the statistics, the hardest Olympiad was that in 1971, followed bythose in 1996, 1993, and 1999. The Olympiad in which the winning team receivedthe lowest score was that in 1977, followed by those in 1960 and 1999.

The selection of the problems consists of several steps. Participant countries sendtheir proposals, which are supposed to be novel, to the IMO organizers. The organiz-ing country does not propose problems. From the received proposals (the longlistedproblems), the problem committee selects a shorter list (the shortlisted problems),which is presented to the IMO jury, consisting of all the team leaders. From theshort-listed problems the jury chooses six problems for the IMO.

Apart from its mathematical and competitive side, the IMO is also a very largesocial event. After their work is done, the students have three days to enjoy eventsand excursions organized by the host country, as well as to interact and socializewith IMO participants from around the world. All this makes for a truly memorableexperience.

1.2 The IMO Compendium

Olympiad problems have been published in many books [97]. However, the remain-ing shortlisted and longlisted problems have not been systematically collected andpublished, and therefore many of them are unknown to mathematicians interested inthis subject. Some partial collections of shortlisted and longlisted problems can befound in the references, though usually only for one year. References [1], [39], [57],[88] contain problems from multiple years. In total, these books cover roughly 50%of the problems found in this book.

The goal of this book is to present, in a single volume, our comprehensive col-lection of problems proposed for the IMO. It consists of all problems selected forthe IMO competitions, shortlisted problems from the 10th IMO and from the 12ththrough 50th IMOs, and longlisted problems from twenty IMOs. We do not haveshortlisted problems from the 9th and the 11th IMOs, and we could not discoverwhether competition problems at those two IMOs were selected from the longlistedproblems or whether there existed shortlisted problems that have not been preserved.Since IMO organizers usually do not distribute longlisted problems to the representa-tives of participant countries, our collection is incomplete. The practice of distribut-

1.2 The IMO Compendium 3

ing these longlists effectively ended in 1989. A selection of problems from the firsteight IMOs has been taken from [88].

The book is organized as follows. For each year, the problems that were given onthe IMO contest are presented, along with the longlisted and/or shortlisted problems,if applicable. We present solutions to all shortlisted problems. The problems appear-ing on the IMOs are solved among the other shortlisted problems. The longlistedproblems have not been provided with solutions, except for the two IMOs held inYugoslavia (for patriotic reasons), since that would have made the book unreason-ably long. This book has thus the added benefit for professors and team coaches ofbeing a suitable book from which to assign problems. For each problem, we indi-cate the country that proposed it with a three-letter code. A complete list of countrycodes and the corresponding countries is given in the appendix. In all shortlists, wealso indicate which problems were selected for the contest. We occasionally makereferences in our solutions to other problems in a straightforward way. After indicat-ing with LL, SL, or IMO whether the problem is from a longlist, shortlist, or contest,we indicate the year of the IMO and then the number of the problem. For example,(SL89-15) refers to the fifteenth problem of the shortlist of 1989.

We also present a rough list of all formulas and theorems not obviously derivablethat were called upon in our proofs. Since we were largely concerned with only thetheorems used in proving the problems of this book, we believe that the list is a goodcompilation of the most useful theorems for IMO problem solving.

The gathering of such a large collection of problems into a book required a mas-sive amount of editing. We reformulated the problems whose original formulationswere not precise or clear. We translated the problems that were not in English. Someof the solutions are taken from the author of the problem or other sources, while oth-ers are original solutions of the authors of this book. Many of the non-original solu-tions were significantly edited before being included. We do not make any guaranteethat the problems in this book fully correspond to the actual shortlisted or longlistedproblems. However, we believe this book to be the closest possible approximation tosuch a list.

2

Basic Concepts and Facts

The following is a list of the most basic concepts and theorems frequently used inthis book. We encourage the reader to become familiar with them and perhaps readup on them further in other literature.

2.1 Algebra

2.1.1 Polynomials

Theorem 2.1. The quadratic equation ax2 +bx+ c = 0 (a,b,c R, a 6= 0) has solu-tions

x1,2 =b

b2 4ac

2a.

The discriminant D of a quadratic equation is defined as D = b24ac. For D < 0 thesolutions are complex and conjugate to each other, for D = 0 the solutions degenerateto one real solution, and for D > 0 the equation has two distinct real solutions.

Definition 2.2. Binomial coefficients(n

k

), n,k N0, k n, are defined as

(ni

)=

n!i!(n i)! .

They satisfy(n

i

)+( n

i1)

=(n+1

i

)for i > 0 and also

(n0

)+(n

1

)+ +

(nn

)= 2n,

(n0

)(n

1

)+ +(1)n

(nn

)= 0,

(n+mk

)= ki=0

(ni

)( mki),(n+r

n

)= rj=0

(n+ j1n1

).

Theorem 2.3 ((Newtons) binomial formula). For x,y C and n N,

(x + y)n =n

i=0

(ni

)xniyi.

Theorem 2.4 (Bzouts theorem). A polynomial P(x) is divisible by the binomialx a (a C) if and only if P(a) = 0.

Springer Science + Business Media, LLC 2011

et al., The IMO Compendium, Problem Books in Mathematics, 5DOI 10.1007/978-1-4419-9854-5_2,

D. Djuki

6 2 Basic Concepts and Facts

Theorem 2.5 (The rational root theorem). If x = p/q is a rational zero of a poly-nomial P(x) = anxn + +a0 with integer coefficients and (p,q) = 1, then p | a0 andq | an.

Theorem 2.6 (The fundamental theorem of algebra). Every nonconstant polyno-mial with coefficients in C has a complex root.

Theorem 2.7 (Eisensteins criterion (extended)). Let P(x) = anxn + + a1x +a0be a polynomial with integer coefficients. If there exist a prime p and an integerk {0,1, . . . ,n1} such that p | a0,a1, . . . ,ak, p ak+1, and p2 a0, then there existsan irreducible factor Q(x) of P(x) whose degree is greater than k. In particular, if pcan be chosen such that k = n1, then P(x) is irreducible.

Definition 2.8. Symmetric polynomials in x1, . . . ,xn are polynomials that do notchange on permuting the variables x1, . . . ,xn. Elementary symmetric polynomialsare k(x1, . . . ,xn) = xi1 xik (the sum is over all k-element subsets {i1, . . . , ik} of{1,2, . . . ,n}).

Theorem 2.9. Every symmetric polynomial in x1, . . . ,xn can be expressed as a poly-nomial in the elementary symmetric polynomials 1, . . . ,n.

Theorem 2.10 (Vites formulas). Let 1, . . . ,n and c1, . . . ,cn be complex numberssuch that

(x1)(x2) (xn) = xn + c1xn1 + c2xn2 + + cn .

Then ck = (1)kk(1, . . . ,n) for k = 1,2, . . . ,n.

Theorem 2.11 (Newtons formulas on symmetric polynomials). Let k = k(x1,. . . , xn) and let sk = xk1 + x

k2 + + xkn, where x1, . . . ,xn are arbitrary complex num-

bers. Then

kk = s1k1 s2k2 + +(1)ksk11 +(1)k1sk .

2.1.2 Recurrence Relations

Definition 2.12. A recurrence relation is a relation that determines the elements of asequence xn, n N0, as a function of previous elements. A recurrence relation of theform

(n k) xn +a1xn1 + +akxnk = 0for constants a1, . . . ,ak is called a linear homogeneous recurrence relation of orderk. We define the characteristic polynomial of the relation as P(x) = xk + a1xk1 + + ak.

Theorem 2.13. Using the notation introduced in the above definition, let P(x) factor-ize as P(x) = (x1)k1(x2)k2 (xr)kr , where 1, . . . ,r are distinct complex

2.1 Algebra 7

numbers and k1, . . . ,kr are positive integers. The general solution of this recurrencerelation is in this case given by

xn = p1(n)n1 + p2(n)n2 + + pr(n)nr ,

where pi is a polynomial of degree less than ki. In particular, if P(x) has k distinctroots, then all pi are constant.

If x0, . . . ,xk1 are set, then the coefficients of the polynomials are uniquely deter-mined.

2.1.3 Inequalities

Theorem 2.14. The squaring function is always positive; i.e., (x R) x2 0. Bysubstituting different expressions for x, many of the inequalities below are obtained.

Theorem 2.15 (Bernoullis inequalities).

1. If n 1 is an integer and x > 1 a real number, then (1 + x)n 1+ nx.2. If > 1 or < 0, then for x > 1, the following inequality holds: (1 + x)

1 +x.

3. If (0,1) then for x > 1 the following inequality holds: (1+ x) 1 + x.

Theorem 2.16 (The mean inequalities). For positive real numbers x1, x2, . . . ,xn itis always the case that QM AM GM HM, where

QM =

x21 + + x2n

n, AM =

x1 + + xnn

,

GM = n

x1 xn, HM =n

1x1

+ + 1xn.

Each of these inequalities becomes an equality if and only if x1 = x2 = = xn.The numbers QM, AM, GM, and HM are respectively called the quadratic mean, thearithmetic mean, the geometric mean, and the harmonic mean of x1,x2, . . . ,xn.

Theorem 2.17 (The general mean inequality). Let x1, . . . ,xn be positive real num-bers. For each p R we define the mean of order p of x1, . . . ,xn by

Mp =

(xp1 + + x

pn

n

)1/p

for p 6= 0, and Mq = limpq Mp for q {,0}. Then

Mp Mq whenever p q.

Remark. In particular, maxxi, QM, AM, GM, HM, and minxi are M, M2, M1, M0,M1, and M respectively.


Theorem 2.18 (CauchySchwarz inequality). Let ai,bi, i = 1,2, . . . ,n, be real num-bers. Then (

n

i=1

aibi

)2(

n

i=1

a2i

)(n

i=1

b2i

).

Equality occurs if and only if there exists c R such that bi = cai for i = 1, . . . ,n.

Theorem 2.19 (Hlders inequality). Let ai,bi, i = 1,2, . . . ,n, be nonnegative realnumbers, and let p,q be positive real numbers such that 1/p+ 1/q = 1. Then

n

i=1

aibi (

n

i=1

api

)1/p( ni=1

bqi

)1/q.

Equality occurs if and only if there exists c R such that bi = cai for i = 1, . . . ,n. TheCauchySchwarz inequality is a special case of Hlders inequality for p = q = 2.

Theorem 2.20 (Minkowskis inequality). Let ai,bi (i = 1,2, . . . ,n) be nonnegativereal numbers and p any real number not smaller than 1. Then

(n

i=1

(ai +bi)p

)1/p(

n

i=1

api

)1/p+

(n

i=1

bpi

)1/p.

For p > 1 equality occurs if and only if there exists c R such that bi = cai fori = 1, . . . ,n. For p = 1 equality occurs in all cases.

Theorem 2.21 (Chebyshevs inequality). Let a1 a2 an and b1 b2 bn be real numbers. Then

nn

i=1

aibi (

n

i=1

ai

)(n

i=1

bi

) n

n

i=1

aibn+1i.

The two inequalities become equalities at the same time when a1 = a2 = = an orb1 = b2 = = bn.

Definition 2.22. A real function f defined on an interval I is convex if f (x+y) f (x)+ f (y) for all x,y I and all , > 0 such that + = 1. A function f issaid to be concave if the opposite inequality holds, i.e., if f is convex.

Theorem 2.23. If f is continuous on an interval I, then f is convex on that intervalif and only if

f

(x + y

2

) f (x)+ f (y)

2for all x,y I.

Theorem 2.24. If f is differentiable, then it is convex if and only if the derivative f

is nondecreasing. Similarly, differentiable function f is concave if and only if f isnonincreasing.

2.1 Algebra 9

Theorem 2.25 (Jensens inequality). If f : I R is a convex function, then theinequality

f (1x1 + +nxn) 1 f (x1)+ +n f (xn)holds for all i 0, 1 + +n = 1, and xi I. For a concave function the oppositeinequality holds.

Theorem 2.26 (Muirheads inequality). Given x1,x2, . . . ,xn R+ and an n-tuplea = (a1, . . . ,an) of positive real numbers, we define

Ta(x1, . . . ,xn) = ya11 yann ,the sum being taken over all permutations y1, . . . ,yn of x1, . . . ,xn. We say that an n-tuple a majorizes an n-tuple b if a1 + + an = b1 + + bn and a1 + + ak b1 + +bk for each k = 1, . . . ,n1. If a nonincreasing n-tuple a majorizes a non-increasing n-tuple b, then the following inequality holds:

Ta(x1, . . . ,xn) Tb(x1, . . . ,xn).

Equality occurs if and only if x1 = x2 = = xn.

Theorem 2.27 (Schurs inequality). Using the notation introduced for Muirheadsinequality,

T+2,0,0(x1,x2,x3)+ T ,,(x1,x2,x3) 2T+,,0(x1,x2,x3),

where R, > 0. Equality occurs if and only if x1 = x2 = x3 or x1 = x2, x3 = 0(and in analogous cases). An equivalent form of the Schurs inequality is

x (x y)(x z)+ y (y x)(y z)+ z(z x)(z y) 0.

2.1.4 Groups and Fields

Definition 2.28. A group is a nonempty set G equipped with a binary operation satisfying the following conditions:

(i) a (b c) = (a b) c for all a,b,c G.(ii) There exists a (unique) identity e G such that ea = a e = a for all a G.

(iii) For each a G there exists a (unique) inverse a1 = b G such that a b =b a = e.

If n Z, we define an as a a a (n times) if n 0, and as (a1)n otherwise.

Definition 2.29. A group G = (G,) is commutative or abelian if ab = ba for alla,b G.

Definition 2.30. A set A generates a group (G,) if every element of G can be ob-tained using powers of the elements of A and the operation . In other words, if A isthe generator of a group G, then every element g G can be written as ai11 ainn ,where a j A and i j Z for every j = 1,2, . . . ,n.


Definition 2.31. The order of an element a G is the smallest n N, if it exists suchthat an = e. If no such n exists then the element a is said to be of infinite order. Theorder of a group is the number of its elements, if it is finite. Each element of a finitegroup has finite order.

Theorem 2.32 (Lagranges theorem). In a finite group, the order of an elementdivides the order of the group.

Definition 2.33. A ring is a nonempty set R equipped with two operations + and such that (R,+) is an abelian group and for any a,b,c R,(i) (a b) c = a (b c);

(ii) (a +b) c = a c + b c and c (a +b) = c a+ c b.A ring is commutative if a b = b a for any a,b R and with identity if there existsa multiplicative identity i R such that i a = a i = a for all a R.

Definition 2.34. A field is a commutative ring with identity in which every elementa other than the additive identity has a multiplicative inverse a1 such that a a1 =a1 a = i.

Theorem 2.35. The following are common examples of groups, rings, and fields:

Groups: (Zn,+), (Zp \{0}, ), (Q,+), (R,+), (R \{0}, ).Rings: (Zn,+, ), (Z,+, ), (Z[x],+, ), (R[x],+, ).Fields: (Zp,+, ), (Q,+, ), (Q(

2),+, ), (R,+, ), (C,+, ).

2.2 Analysis

Definition 2.36. A sequence {an}n=1 of real numbers has a limit a = limn an(also denoted by an a) if

( > 0)(n N)(n n) |an a|< .

A function f : (a,b) R has a limit y = limxc f (x) if

( > 0)( > 0)(x (a,b)) 0 < |x c|< | f (x) y| < .

Definition 2.37. A sequence {xn} converges to x R if limn xn = x. A seriesn=1 xn converges to s R if and only if limm mn=1 xn = s. A sequence or seriesthat does not converge is said to diverge.

Theorem 2.38. A sequence {an} of real numbers is convergent if it is monotonic andbounded.

2.2 Analysis 11

Definition 2.39. A function f is continuous on [a,b] if the following three relationshold:

limxx0

f (x) = f (x0), for every x0 (a,b),

limxa+

f (x) = f (a),

and limxb

f (x) = f (b).

Definition 2.40. A function f : (a,b) R is differentiable at a point x0 (a,b) ifthe following limit exists:

f (x0) = limxx0

f (x) f (x0)x x0

.

A function is differentiable on (a,b) if it is differentiable at every x0 (a,b). Thefunction f is called the derivative of f . We similarly define the second derivative f

as the derivative of f , and so on.

Theorem 2.41. A differentiable function is also continuous. If f and g are differen-tiable, then f g, f + g (, R), f g, 1/ f (if f 6= 0), f1 (if well defined) arealso differentiable. It holds that ( f + g) = f +g, ( f g) = f g+ f g, ( f g) =( f g) g, (1/ f ) = f / f 2, ( f/g) = ( f g f g)/g2, ( f1) = 1/( f f1).

Theorem 2.42. The following are derivatives of some elementary functions (a de-notes a real constant): (xa) = axa1, (lnx) = 1/x, (ax) = ax lna, (sinx) = cosx,(cosx) = sinx.

Theorem 2.43 (Fermats theorem). Let f : [a,b] R be a continuous function thatis differentiable at every point of (a,b). The function f attains its maximum andminimum in [a,b]. If x0 (a,b) is a number at which the extremum is attained (i.e.,f (x0) is the maximum or minimum), then f (x0) = 0.

Theorem 2.44 (Rolles theorem). Let f (x) be a continuous function defined on[a,b], where a,b R, a < b, and f (a) = f (b). If f is differentiable in (a,b), thenthere exists c (a,b) such that f (c) = 0.

Definition 2.45. Differentiable functions f1, f2, . . . , fk defined on an open subset Dof Rn are independent if there is no nonzero differentiable function F : Rk R suchthat F( f1, . . . , fk) is identically zero on some open subset of D.

Theorem 2.46. Functions f1, . . . , fk : D R are independent if and only if the knmatrix [ fi/x j]i, j is of rank k, i.e., when its k rows are linearly independent at somepoint.


Theorem 2.47 (Lagrange multipliers). Let D be an open subset of Rn and f , f1, f2,. . . , fk : D R independent differentiable functions. Assume that a point a in D isan extremum of the function f within the set of points in D for which f1 = f2 = =fk = 0. Then there exist real numbers 1, . . . ,k (so-called Lagrange multipliers)such that a is a stationary point of the function F = f + 1 f1 + +k fk, i.e., suchthat all partial derivatives of F at a are zero.

Definition 2.48. Let f be a real function defined on [a,b] and let a = x0 x1 xn = b and k [xk1,xk]. The sum S = nk=1(xk xk1) f (k) is called a Darbouxsum. If I = lim0 S exists (where = maxk(xk xk1)), we say that f is integrableand that I is its integral. Every continuous function is integrable on a finite interval.

2.3 Geometry

2.3.1 Triangle Geometry

Definition 2.49. The orthocenter of a triangle is the common point of its three alti-tudes.

Definition 2.50. The circumcenter of a triangle is the center of its circumscribedcircle (i.e., circumcircle). It is the common point of the perpendicular bisectors ofthe sides of the triangle.

Definition 2.51. The incenter of a triangle is the center of its inscribed circle (i.e.,incircle). It is the common point of the internal bisectors of its angles.

Definition 2.52. The centroid of a triangle (median point) is the common point ofits medians.

Theorem 2.53. The orthocenter, circumcenter, incenter, and centroid are well de-fined (and unique) for every nondegenerate triangle.

Theorem 2.54 (Eulers line). The orthocenter H, centroid G, and circumcenter Oof an arbitrary triangle lie on a line and satisfy

HG = 2

GO.

Theorem 2.55 (The nine-point circle). The feet of the altitudes from A, B, C and themidpoints of AB, BC, CA, AH, BH, CH lie on a circle.

Theorem 2.56 (Feuerbachs theorem). The nine-point circle of a triangle is tangentto the incircle and all three excircles of the triangle.

Theorem 2.57 (Torricellis point). Given a triangle ABC, let ABC, ABC,and ABC be equilateral triangles constructed outward. Then AA, BB, CC inter-sect in one point.

Definition 2.58. Let ABC be a triangle, P a point, and X , Y , Z respectively the feet ofthe perpendiculars from P to BC, AC, AB. Triangle XYZ is called the pedal triangleof ABC corresponding to point P.

2.3 Geometry 13

Theorem 2.59 (Simsons line). The pedal triangle XYZ is degenerate, i.e., X, Y , Zare collinear, if and only if P lies on the circumcircle of ABC. Points X, Y , Z are inthis case said to lie on Simsons line.

Theorem 2.60. If M is a point on the circumcircle of ABC with orthocenter H,then the Simsons line corresponding to M bisects the segment MH.

Theorem 2.61 (Carnots theorem). The perpendiculars from X ,Y,Z to BC,CA,ABrespectively are concurrent if and only if

BX 2 XC2 +CY 2 YA2 + AZ2 ZB2 = 0.

Theorem 2.62 (Desarguess theorem). Let A1B1C1 and A2B2C2 be two triangles.The lines A1A2, B1B2, C1C2 are concurrent or mutually parallel if and only if thepoints A = B1C1 B2C2, B = C1A1 C2A2, and C = A1B1 A2B2 are collinear.

Definition 2.63. Given a point C in the plane and a real number r, a homothety withcenter C and coefficient r is a mapping of the plane that sends each point A to the

point A such thatCA = k

CA.

Theorem 2.64. Let k1, k2, and k3 be three circles. Then the three external similitudecenters of these three circles are collinear (the external similitude center is the centerof the homothety with positive coefficient that maps one circle to the other). Similarly,two internal similitude centers are collinear with the third external similitude center.

All variants of the previous theorem can be directly obtained from the Desar-guess theorem applied to the following two triangles: the first triangle is determinedby the centers of k1, k2, k3, while the second triangle is determined by the points oftangency of an appropriately chosen circle that is tangent to all three of k1, k2, k3.

2.3.2 Vectors in Geometry

Definition 2.65. For any two vectors a ,b in space, we define the scalar product(also known as dot product) of a and b as a b = |a ||b |cos , and the vectorproduct (also known as cross product) as a b = p , where = (a ,b ) andp is the vector with |p | = |a ||b ||sin| perpendicular to the plane determined bya and b such that the triple of vectors a ,b ,p is positively oriented (note thatif a and b are collinear, then a b = 0 ). Both these products are linear withrespect to both factors. The scalar product is commutative, while the vector product isanticommutative, i.e., a b = b a . We also define the mixed vector productof three vectors a ,b ,c as [a ,b ,c ] = (a b ) c .Remark. The scalar product of vectors a and b is often denoted by a ,b .

Theorem 2.66 (Thales theorem). Let lines AA and BB intersect in a point O, A 6=O 6= B. Then AB AB

OAOA

=OBOB

(Hereab

denotes the ratio of two nonzero

collinear vectors).


Theorem 2.67 (Cevas theorem). Let ABC be a triangle and X, Y , Z points on linesBC, CA, AB respectively, distinct from A,B,C. Then the lines AX, BY , CZ are con-current if and only if

BXXC

CYYA

AZZB

= 1, or equivalently,sinBAXsinXAC

sinCBYsinYBA

sinACZsinZCB

= 1

(the last expression being called the trigonometric form of Cevas theorem).

Theorem 2.68 (Menelauss theorem). Using the notation introduced for Cevastheorem, points X ,Y,Z are collinear if and only if

BXXC

CYYA

AZZB

= 1.

Theorem 2.69 (Stewarts theorem). If D is an arbitrary point on the line BC, then

AD2 =DCBC

BD2 +BDBC

CD2 BD DC.

Specifically, if D is the midpoint of BC, then 4AD2 = 2AB2 + 2AC2BC2.

2.3.3 Barycenters

Definition 2.70. A mass point (A,m) is a point A that is assigned a mass m > 0.

Definition 2.71. The center of mass (barycenter) of the set of mass points (Ai,mi),i = 1,2, . . . ,n, is the point T such that i mi

TAi =

0 .

Theorem 2.72 (Leibnizs theorem). Let T be the mass center of the set of masspoints {(Ai,mi) | i = 1,2, . . . ,n} of total mass m = m1 + + mn, and let X be anarbitrary point. Then

n

i=1

miXA2i =

n

i=1

miTA2i + mXT

2.

Specifically, if T is the centroid of ABC and X an arbitrary point, then

AX2 + BX2 +CX2 = AT 2 +BT 2 +CT 2 +3XT2 .

2.3.4 Quadrilaterals

Theorem 2.73. A quadrilateral ABCD is cyclic (i.e., there exists a circumcircle ofABCD) if and only if ACB = ADB and if and only if ADC +ABC = 180.

2.3 Geometry 15

Theorem 2.74 (Ptolemys theorem). A convex quadrilateral ABCD is cyclic if andonly if

AC BD = AB CD + AD BC.For an arbitrary quadrilateral ABCD we have Ptolemys inequality (see 2.3.7, Geo-metric Inequalities).

Theorem 2.75 (Caseys theorem). Let k1, k2, k3, and k4 be four circles that all toucha given circle k. Let ti j be the length of a segment determined by an external commontangent of circles ki and k j (i, j {1,2,3,4}) if both ki and k j touch k internally, orboth touch k externally. Otherwise, ti j is set to be the internal common tangent. Thenone of the products t12t34, t13t24, and t14t23 is the sum of the other two.

Some of the circles k1, k2, k3, k4 may be degenerate, i.e., of 0 radius, and thusreduced to being points. In particular, for three points A, B, C on a circle k and acircle k touching k at a point on the arc of AC not containing B, we have AC b =AB c+ a BC, where a, b, and c are the lengths of the tangent segments from pointsA, B, and C to k. Ptolemys theorem is a special case of Caseys theorem when allfour circles are degenerate.

Theorem 2.76. A convex quadrilateral ABCD is tangent (i.e., there exists an incircleof ABCD) if and only if

AB +CD = BC +DA.

Theorem 2.77. For arbitrary points A,B,C,D in space, AC BD if and only if

AB2 +CD2 = BC2 +DA2.

Theorem 2.78 (Newtons theorem). Let ABCD be a quadrilateral, ADBC = E,and ABDC = F (such points A,B,C,D,E,F form a complete quadrilateral). Thenthe midpoints of AC, BD, and EF are collinear. If ABCD is tangent, then the incenteralso lies on this line.

Theorem 2.79 (Brocards theorem). Let ABCD be a quadrilateral inscribed in acircle with center O, and let P = ABCD, Q = ADBC, R = ACBD. Then O isthe orthocenter of PQR.

2.3.5 Circle Geometry

Theorem 2.80 (Pascals theorem). If A1,A2,A3,B1,B2,B3 are distinct points on aconic (e.g., circle), then points X1 = A2B3 A3B2, X2 = A1B3 A3B1, and X3 =A1B2 A2B1 are collinear. The special result when consists of two lines is calledPappuss theorem.

Theorem 2.81 (Brianchons theorem). Let ABCDEF be a convex hexagon. If aconic (e.g., circle) can be inscribed in ABCDEF, then AD, BE, and CF meet in apoint.


Theorem 2.82 (The butterfly theorem). Let AB be a chord of a circle k and C itsmidpoint. Let p and q be two different lines through C that, respectively, intersect kon one side of AB in P and Q and on the other in P and Q. Let E and F respectivelybe the intersections of PQ and PQ with AB. Then it follows that CE = CF.

Definition 2.83. The power of a point X with respect to a circle k(O,r) is definedby P(X) = OX2 r2. For an arbitrary line l through X that intersects k at A and B(A = B when l is a tangent), it follows that P(X) =

XA XB.

Definition 2.84. The radical axis of two circles is the locus of points that haveequal powers with respect to both circles. The radical axis of circles k1(O1,r1) andk2(O2,r2) is a line perpendicular to O1O2. The radical axes of three distinct circlesare concurrent or mutually parallel. If concurrent, the intersection of the three axesis called the radical center.

Definition 2.85. The pole of a line l 6 O with respect to a circle k(O,r) is a point Aon the other side of l from O such that OA l and d(O, l) OA = r2. In particular, if lintersects k in two points, its pole will be the intersection of the tangents to k at thesetwo points.

Definition 2.86. The polar of the point A from the previous definition is the line l.In particular, if A is a point outside k and AM, AN are tangents to k (M,N k), thenMN is the polar of A.Poles and polars are generally defined in a similar way with respect to arbitrarynondegenerate conics.

Theorem 2.87. If A belongs to the polar of B, then B belongs to the polar of A.

2.3.6 Inversion

Definition 2.88. An inversion of the plane about the circle k(O,r) (which belongsto ) is a transformation of the set \{O} onto itself such that every point P istransformed into a point P on the ray (OP such that OP OP = r2. In the followingstatements we implicitly assume exclusion of O.

Theorem 2.89. The fixed points of an inversion about a circle k are on the circle k.The inside of k is transformed into the outside and vice versa.

Theorem 2.90. If A, B transform into A, B after an inversion about a circle k, thenOAB = OBA, and also ABBA is cyclic and perpendicular to k. A circle perpen-dicular to k transforms into itself. Inversion preserves angles between continuouscurves (which includes lines and circles).

Theorem 2.91. An inversion transforms lines not containing O into circles contain-ing O, lines containing O into themselves, circles not containing O into circles notcontaining O, circles containing O into lines not containing O.

2.3 Geometry 17

2.3.7 Geometric Inequalities

Theorem 2.92 (The triangle inequality). For any three points A, B, C, AB +BC AC. Equality occurs when A, B, C are collinear and B is between A and C. In thesequel we will use B(A,B,C) to emphasize that B is between A and C.

Theorem 2.93 (Ptolemys inequality). For any four points A, B, C, D,

AC BD AB CD +AD BC.

Theorem 2.94 (The parallelogram inequality). For any four points A, B, C, D,

AB2 +BC2 +CD2 + DA2 AC2 +BD2.

Equality occurs if and only if ABCD is a parallelogram.

Theorem 2.95. For a given triangle ABC the point X for which AX +BX +CX isminimal is Toricellis point when all angles of ABC are less than or equal to 120,and is the vertex of the obtuse angle otherwise. The point X2 for which AX22 +BX

22 +

CX22 is minimal is the centroid (see Leibnizs theorem).

Theorem 2.96 (The ErdosMordell inequality). Let P be a point in the interior ofABC and X ,Y,Z projections of P onto BC,AC,AB, respectively. Then

PA +PB +PC 2(PX + PY + PZ).

Equality holds if and only if ABC is equilateral and P is its center.

2.3.8 Trigonometry

Definition 2.97. The trigonometric circle is the unit circle centered at the origin O ofa coordinate plane. Let A be the point (1,0) and P(x,y) a point on the trigonometriccircle such that AOP = . We define sin = y, cos = x, tan = y/x, and cot =x/y.

Theorem 2.98. The functions sin and cos are periodic with period 2 . The func-tions tan and cot are periodic with period . The following simple identities hold:sin2 x+cos2 x = 1, sin0 = sin = 0, sin(x) =sinx, cos(x) = cosx, sin(/2) =1, sin(/4) = 1/

2, sin(/6) = 1/2, cosx = sin(/2 x). From these identities

other identities can be easily derived.

Theorem 2.99. Additive formulas for trigonometric functions:

sin( ) = sin cos cos sin , cos( ) = cos cos sin sin ,tan( ) = tantan1tan tan , cot( ) =

cot cot1cotcot .


Theorem 2.100. Formulas for trigonometric functions of 2x and 3x:

sin 2x = 2sinxcosx, sin3x = 3sinx 4sin3 x,cos2x = 2cos2 x1, cos3x = 4cos3 x3cosx,tan2x = 2 tanx

1tan2 x , tan3x =3 tanxtan3 x

13 tan2 x .

Theorem 2.101. For any x R, sinx = 2t1+t2

and cosx = 1t2

1+t2, where t = tan x2 .

Theorem 2.102. Transformations from product to sum:

2cos cos = cos( + )+ cos( ),2sin cos = sin( + )+ sin( ),2sin sin = cos( ) cos( + ).

Theorem 2.103. The angles , , of a triangle satisfy

cos2 + cos2 + cos2 +2cos cos cos = 1,tan + tan + tan = tan tan tan.

Theorem 2.104 (De Moivres formula). If i2 = 1, then

(cosx + isinx)n = cosnx+ isinnx.

2.3.9 Formulas in Geometry

Theorem 2.105 (Herons formula). The area of a triangle ABC with sides a,b,cand semiperimeter s is given by

S =

s(sa)(sb)(s c) = 14

2a2b2 +2a2c2 +2b2c2 a4 b4 c4.

Theorem 2.106 (The law of sines). The sides a,b,c and angles , , of a triangleABC satisfy

asin

=b

sin=

csin

= 2R,

where R is the circumradius of ABC.

Theorem 2.107 (The law of cosines). The sides and angles of ABC satisfy

c2 = a2 +b22abcos.

Theorem 2.108. The circumradius R and inradius r of a triangle ABC satisfy R =abc4S and r =

2Sa+b+c = R(cos +cos +cos 1). If x,y,z denote the distances of the

circumcenter in an acute triangle to the sides, then x + y + z = R + r.

Theorem 2.109 (Eulers formula). If O and I are the circumcenter and incenter ofABC, then OI2 = R(R2r), where R and r are respectively the circumradius andthe inradius of ABC. Consequently, R 2r.

2.4 Number Theory 19

Theorem 2.110. If a, b, c, d are lengths of the sides of a convex quadrilateral, p itssemiperimeter, and and two non-adjacent angles of the quadrilateral, then itsarea S is given by

S =

(pa)(pb)(p c)(pd)abcd cos2 +

2.

If the quadrilateral is cyclic, the above formula reduces to

S =

(pa)(p b)(p c)(p d).

Theorem 2.111 (Eulers theorem for pedal triangles). Let X ,Y,Z be the feet ofthe perpendiculars from a point P to the sides of a triangle ABC. Let O denote thecircumcenter and R the circumradius of ABC. Then

SXY Z =14

1OP2

R2

SABC .

Moreover, SXY Z = 0 if and only if P lies on the circumcircle of ABC (see Simsonsline).

Theorem 2.112. If a = (a1,a2,a3),b = (b1,b2,b3),

c = (c1,c2,c3) are three vec-tors in coordinate space, then

a b = a1b1 + a2b2 + a3b3, a b = (a1b2 a2b1,a2b3 a3b2,a3b1 a1b3),

[a ,b ,c ] = det

a1 a2 a3b1 b2 b3c1 c2 c3

.

Here detM denotes the determinant of the square matrix M.

Theorem 2.113. The area of a triangle ABC and the volume of a tetrahedron ABCD

are equal to 12 |ABAC| and 16

[AB,

AC,

AD], respectively.

Theorem 2.114 (Cavalieris principle). If the sections of two solids by the sameplane always have equal area, then the volumes of the two solids are equal.

2.4 Number Theory

2.4.1 Divisibility and Congruences

Definition 2.115. The greatest common divisor (a,b) = gcd(a,b) of a,b N is thelargest positive integer that divides both a and b. Positive integers a and b are coprimeor relatively prime if (a,b) = 1. The least common multiple [a,b] = lcm(a,b) ofa,b N is the smallest positive integer that is divisible by both a and b. It holdsthat [a,b](a,b) = ab. The above concepts are easily generalized to more than twonumbers; i.e., we also define (a1,a2, . . . ,an) and [a1,a2, . . . ,an].


Theorem 2.116 (Euclidean algorithm). Since (a,b) = (|a b|,a) = (|a b|,b), itfollows that starting from positive integers a and b one eventually obtains (a,b) byrepeatedly replacing a and b with |a b| and min{a,b} until the two numbers areequal. The algorithm can be generalized to more than two numbers.

Theorem 2.117 (Corollary to Euclidean algorithm). For each a,b N there existx,y Z such that ax+by = (a,b). The number (a,b) is the smallest positive numberfor which such x and y can be found.

Theorem 2.118 (Second corollary to Euclids algorithm). For a,m,n N and a >1 it follows that (am 1,an1) = a(m,n) 1.

Theorem 2.119 (Fundamental theorem of arithmetic). Every positive integer canbe uniquely represented as a product of primes, up to their order.

Theorem 2.120. The fundamental theorem of arithmetic also holds in some otherrings, such as Z[i] = {a+bi | a,bZ}, Z[

2], Z[

2], Z[ ] (where is a complex

third root of 1). In these cases, the factorization into primes is unique up to the orderand divisors of 1.

Definition 2.121. Integers a,b are congruent modulo nN if n | ab. We then writea b (mod n).

Theorem 2.122 (Chinese remainder theorem). If m1,m2, . . . ,mk are positive in-tegers pairwise relatively prime and a1, . . . ,ak, c1, . . . ,ck are integers such that(ai,mi) = 1 (i = 1, . . . ,k), then the system of congruences

aix ci (mod mi), i = 1,2, . . . ,k ,

has a unique solution modulo m1m2 mk.

2.4.2 Exponential Congruences

Theorem 2.123 (Wilsons theorem). If p is a prime, then p | (p1)! +1.

Theorem 2.124 (Fermats (little) theorem). Let p be a prime number and a aninteger with (a, p) = 1. Then ap1 1 (mod p). This theorem is a special case ofEulers theorem.

Definition 2.125. Eulers function (n) is defined for nN as the number of positiveintegers less than or equal to n and coprime to n. It holds that

(n) = n(

1 1p1

) (

1 1pk

),

where n = p11 pkk is the factorization of n into primes.

Theorem 2.126 (Eulers theorem). Let n be a natural number and a an integer with(a,n) = 1. Then a(n) 1 (mod n).

2.4 Number Theory 21

Theorem 2.127 (Existence of primitive roots). Let p be a prime number. Thereexists g {1,2, . . . , p 1} (called a primitive root modulo p) such that the set{1,g,g2, . . . ,gp2} is equal to {1, 2, . . . , p1} modulo p.

Definition 2.128. Let p be a prime and a nonnegative integer. We say that p isthe exact power of p that divides an integer a (and the exact exponent) if p | aand p+1 a.

Theorem 2.129. Let a and n be positive integers and p an odd prime. If p ( N)is the exact power of p that divides a 1, then for any integer 0, p+ | an 1if and only if p | n. (See (SL97-14).)

A similar statement holds for p = 2. If 2 ( N) is the exact power of 2 thatdivides a2 1, then for any integer 0, 2+ | an 1 if and only if 2+1 | n. (See(SL89-27).)

2.4.3 Quadratic Diophantine Equations

Theorem 2.130. The solutions of a2+b2 = c2 in integers are given by a = t(m2n2),b = 2tmn, c = t(m2 + n2) (provided that b is even), where t,m,n Z. The triples(a,b,c) are called Pythagorean (or primitive Pythagorean if gcd(a,b,c) = 1).

Definition 2.131. Given D N that is not a perfect square, a Pells equation is anequation of the form x2 Dy2 = 1, where x,y Z.

Theorem 2.132. If (x0,y0) is the least (nontrivial) solution in N of the Pells equationx2 Dy2 = 1, then all the nontrivial integer solutions (x,y) are given by x+ y

D =

(x0 + y0

D)n, where n Z.

Definition 2.133. An integer a is a quadratic residue modulo a prime p if there existsx Z such that x2 a (mod p). Otherwise, a is a quadratic nonresidue modulo p.

Definition 2.134. The Legendre symbol for an integer a and a prime p is defined by

(ap

)=

1 if a is a quadratic residue mod p and p a;0 if p | a;1 otherwise.

Clearly(

ap

)=(

a+pp

)and

(a2p

)= 1 if p a. The Legendre symbol is multiplicative,

i.e.,(

ap

)(bp

)=(

abp

).

Theorem 2.135 (Eulers criterion). For each odd prime p and integer a not divisible

by p, ap1

2 (

ap

)(mod p).

Theorem 2.136. For a prime p > 3,(1p

),(

2p

), and

(3p

)are equal to 1 if and

only if p 1 (mod 4), p 1 (mod 8) and p 1 (mod 6), respectively.


Theorem 2.137 (Gausss reciprocity law). For any two distinct odd primes p andq, we have that (

pq

)(qp

)= (1)

p12

q12 .

Definition 2.138. Jacobi symbol for an integer a and an odd positive integer b isdefined as (a

b

)=

(ap1

)1 (

apk

)k,

where b = p11 pkk is the factorization of b into primes.

Theorem 2.139. If(

ab

)= 1, then a is a quadratic nonresidue modulo b, but the

converse is false. All the above identities for Legendre symbols except Eulers crite-rion remain true for Jacobi symbols.

2.4.4 Farey Sequences

Definition 2.140. For any positive integer n, the Farey sequence Fn is the sequenceof rational numbers a/b with 0 a b n and (a,b) = 1 arranged in increasingorder. For instance, F3 = { 01 , 13 , 12 , 23 , 11}.

Theorem 2.141. If p1/q1, p2/q2, and p3/q3 are three successive terms in a Fareysequence, then

p2q1 p1q2 = 1 andp1 + p3q1 + q3

=p2q2

.

2.5 Combinatorics

2.5.1 Counting of Objects

Many combinatorial problems involving the counting of objects satisfying a givenset of properties can be properly reduced to an application of one of the followingconcepts.

Definition 2.142. A variation of order n over k is a 1to1 mapping of {1,2, . . . ,k}into {1,2, . . . ,n}. For a given n and k, where n k, the number of different variationsis V kn =

n!(nk)! .

Definition 2.143. A variation with repetition of order n over k is an arbitrary map-ping of {1,2, . . . ,k} into {1,2, . . . ,n}. For a given n and k the number of differentvariations with repetition is V

kn = k

n.

Definition 2.144. A permutation of order n is a bijection of {1,2, . . . ,n} into itself (aspecial case of variation for k = n). For a given n the number of different permutationsis Pn = n!.

2.5 Combinatorics 23

Definition 2.145. A combination of order n over k is a k-element subset of {1, 2, . . . ,n}. For a given n and k the number of different combinations is Ckn =

(nk

).

Definition 2.146. A permutation with repetition of order n is a bijection of {1, 2, . . . ,n} into a multiset of n elements. A multiset is defined to be a set in which certainelements are deemed mutually indistinguishable (for example, as in {1,1,2,3}).

If {k1,k2, . . . ,ks} denotes the set of distinct elements in a multiset and the ele-ment ki appears i times in the multiset, then number of different permutations withrepetition is Pn,1,...,s =

n!1!2!s! . A combination is a special case of permutation

with repetition for a multiset with two different elements.

Theorem 2.147 (The pigeonhole principle). If a set of nk + 1 different elements ispartitioned into n mutually disjoint subsets, then at least one subset will contain atleast k + 1 elements.

Theorem 2.148 (The inclusionexclusion principle). Let S1,S2, . . . ,Sn be a familyof subsets of the set S. The number of elements of S contained in none of the subsetsis given by the formula

|S\(S1 Sn)| = |S|n

k=1

1i1


Theorem 2.154 (Halls marriage theorem). Let W, M be a partition of the set ofvertices of a bipartite graph. There exists a marriage f : W M if and only if forevery U W the number |U | is not greater than the total number of neighbors of Uinside M.

Definition 2.155. The degree d(x) of a vertex x is the number of times x is the end-point of an edge (thus, self-connecting edges are counted twice for correspondingvertices). An isolated vertex is one with degree 0.

Theorem 2.156. For a graph G = (V,E) the following identity holds:

xV

d(x) = 2|E|.

As a consequence, the number of vertices of odd degree is even.

Definition 2.157. A trajectory (path) of a graph is a finite sequence of vertices, eachconnected to the previous one. The length of a trajectory is the number of edgesthrough which it passes. A circuit is a path that ends in the starting vertex. A cycle isa circuit in which no vertex appears more than once (except the initial/final vertex).

A graph is connected if there exists a trajectory between any two vertices.

Definition 2.158. A subgraph G = (V ,E ) of a graph G = (V,E) is a graph such thatV V and E contains exactly the edges of E connecting points in V . A connectedcomponent of a graph is a connected subgraph such that no vertex of the subgraph isconnected with any vertex outside of the subgraph.

Definition 2.159. A tree is a connected graph that contains no cycles.

Theorem 2.160. A tree with n vertices has exactly n1 edges and at least two ver-tices of degree 1.

Definition 2.161. An Euler path is a path in which each edge appears exactly once.Likewise, an Euler circuit is an Euler path that is also a circuit.

Theorem 2.162. The following conditions are necessary and sufficient for a finiteconnected graph G to have an Euler path:

The graph contains an Euler circuit if and only if each vertex has even degree. The graph contains an Euler path if and only if the number of vertices of odd

degree is either 0 or 2 (in the latter case the path starts and ends in the two oddvertices).

Definition 2.163. A Hamiltonian circuit is a circuit that contains each vertex of Gexactly once (trivially, it is also a cycle).

A simple rule to determine whether a graph contains a Hamiltonian circuit hasnot yet been discovered.

2.5 Combinatorics 25

Theorem 2.164 (Ores theorem). Let G be a graph with n vertices. If the sum of thedegrees of any two nonadjacent vertices in G is greater than or equal to n, then Ghas a Hamiltonian circuit.

Theorem 2.165 (Ramseys theorem). Let r 1 and q1,q2, . . . ,qs r. There existsa minimal positive integer N(q1,q2, . . . ,qs;r) such that for n N, if all subgraphsKr of Kn are partitioned into s different sets, labeled A1,A2 . . . ,As, then for some ithere exists a complete subgraph Kqi whose subgraphs Kr all belong to Ai. For r = 2this corresponds to coloring the edges of Kn with s different colors and looking for amonochromatic subgraph Kqi in color i.

Theorem 2.166. N(p,q;r) N(N(p1,q;r),N(p,q 1;r);r1)+ 1, and in par-ticular, N(p,q;2) N(p1,q;2)+ N(p,q1;2).

The following values of N are known: N(p,q;1) = p + q 1, N(2, p;2) = p,N(3,3;2) = 6, N(3,4;2) = 9, N(3,5;2) = 14, N(3,6;2) = 18, N(3,7;2) = 23,N(3,8;2) = 28, N(3,9;2) = 36, N(4,4;2) = 18, N(4,5;2) = 25.

Theorem 2.167 (Turns theorem). If a simple graph on n = t(p 1)+ r vertices(0 r < p 1) has more than f (n, p) = (p2)n

2r(p1r)2(p1) edges, then it contains Kp

as a subgraph. The graph containing f (n, p) edges that does not contain Kp is thecomplete multipartite graph with r parts with t +1 vertices, and p1 r parts witht vertices.

Definition 2.168. A planar graph is one that can be embedded in a plane such thatits vertices are represented by points and its edges by lines (not necessarily straight)connecting the vertices such that no two edges intersect each other.

Theorem 2.169. A planar graph with n vertices has at most 3n6 edges.

Theorem 2.170 (Kuratowskis theorem). Graphs K5 and K3,3 are not planar. Everynonplanar graph contains a subgraph that can be obtained from one of these twographs by a subdivison of its edges.

Theorem 2.171 (Eulers formula). For a given convex polyhedron let E be thenumber of its edges, F the number of faces, and V the number of vertices. ThenE + 2 = F +V. The same formula holds for a connected planar graph (F is in thiscase equal to the number of planar regions).

3

Problems

3.1 The First IMOBucharestBrasov, Romania, July 2331, 1959

3.1.1 Contest Problems

First Day

1. (POL) For every integer n prove that the fraction 21n+414n+3 cannot be reduced anyfurther.

2. (ROU) For which real numbers x do the following equations hold:

(a)

x +

2x 1+

x

2x1 =

2 ,

(b)

x +

2x 1+

x

2x1 = 1 ,(c)

x +

2x 1+

x

2x1 = 2 ?

3. (HUN) Let x be an angle and let the real numbers a, b, c, cosx satisfy thefollowing equation:

acos2 x +bcosx + c = 0 .

Write the analogous quadratic equation for a, b, c, cos2x. Compare the givenand the obtained equality for a = 4, b = 2, c = 1.

Second Day

4. (HUN) Construct a right-angled triangle whose hypotenuse c is given if it isknown that the median from the right angle equals the geometric mean of theremaining two sides of the triangle.

5. (ROU) A segment AB is given and on it a point M. On the same side ofAB squares AMKD and BMFE are constructed. The circumcircles of the twosquares, whose centers are P and Q, intersect in M and another point N.

(a) Prove that lines FA and BC intersect at N.

Springer Science + Business Media, LLC 2011

et al., The IMO Compendium, Problem Books in Mathematics, 27DOI 10.1007/978-1-4419-9854-5_3,

D. Djuki

28 3 Problems

(b) Prove that all such constructed lines MN pass through the same point S,regardless of the selection of M.

(c) Find the locus of the midpoints of all segments PQ, as M varies along thesegment AB.

6. (CZS) Let and be two planes intersecting at a line p. In a point A is givenand in a point C is given, neither of which lies on p. Construct B in and Din such that ABCD is an equilateral trapezoid, AB CD, in which a circle canbe inscribed.

3.2 IMO 1960 29

3.2 The Second IMOBucharestSinaia, Romania, July 1825, 1960

3.2.1 Contest Problems

First Day

1. (BGR) Find all the three-digit numbers for which one obtains, when dividingthe number by 11, t

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